I decided to try and analyse the lap times and reevaluate my view on Rosberg's strategy. Here are the lap times of Rosberg around the time of his first stop:
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Rosberg Hamilton Gap
17 1:41.455 17 1:41.802 0.550
18 1:42.587 18 1:42.621 0.516
19 1:43.965 19 P 1:46.545 -2.064
20 1:41.442 20 1:59.311 -19.933
21 P 1:44.439 21 1:39.765 -15.259
22 2:01.019 22 1:39.666 6.094
23 1:39.886 23 1:40.357 5.623
24 1:40.178 24 1:39.743 6.058
Taken from www.fia.com
Rosberg's and Hamiltons' lap 19 should give us a pretty good indication on how much you loose on the inlap as a result of slowing down in the pitlane. It seems to be roughly 2.5 seconds. Rosberg's lap 20 and 21 should give a similar indication, except that his tires were probably falling off lap by lap. At the end of a stint you will use up your tires and the performance will often drop quite rapidly. His lap time went up by 3 seconds on his inlap, so maybe he lost the extra 0.5 seconds as a result of the tires being used up. Looking at other drivers this seems to fit quite well, so I think 2.5 seconds is a reasonable estimate on the time lost in pit lane on the inlap. In that case we can compare Rosberg' lap 21 to Hamilton's lap 21 and we see that he looses 4.674 s, which is then 2.174 s more than the 2.5 seconds resulting from the inlap.
At the end of lap 22, Hamilton's lead was up to 6.094 s. If we subtract the 2.174 seconds that Rosberg lost on lap 21, we are left with 3.920 s that Rosberg lost on laps 20 and 22. Hamilton's outlap on softs was 1:59.311 and Rosberg's outlap on mediums was 2:01.019. This is a difference of 1.708 s, which may well be the result of the medium tires taking longer to heat up. If we subtract the 1.708 s from 3.920 s we are left with 2.212 s. Of course, comparing the outlaps directly isn't totally accurate, as the fuel load decreases and the track rubbers in (maybe not that significant from lap 20 to lap 22). So probably Rosberg a bit lost more on the outlap than 1.708 s and less than 2.212 s on lap 20. The fuel load difference should be about 3.5 kg based on a 100 kg fuel load for 57 laps. This is maybe worth a tenth of second.
So Phil is right, the outlap accounts for, let's say almost 2 seconds of the gap between Hamilton and Rosberg after 22 laps, which means that he only lost around 2 seconds or slightly less per lap by staying out. The 2 seconds he lost on warming up the medium tires can be seen as an investement that he would get back on the final stint.
For the record, the time Hamilton spent in the pit lane was 24.687 s and the time Rosberg spent in the pit lane was 24.851 s, so there was no significant difference there. 0.164 s is not really significant in this estimate. By the way, I chose to include all decimals for accountability.
Having said this, I still find it unlikely that Rosberg's strategy was based on giving him the best opportunities to beat Hamilton. There must be a reason why Hamilton didn't run mediums on his second stint and I believe that the reason most likely is that soft-soft-medium was the fastest strategy. It may be that they wanted to gamble on a safety car for Rosberg, but seing how close Rosberg came to pass Hamilton at the end of the first stint, I can't help but thinking that he would have been better off running softs on the second stint as well and force Hamilton to use his tyres slightly more than he needed to himself, and apply the pressure at the end of the stint. However, I see a very good reason why the Mercedes team would want to avoid close wheel to wheel racing between their drivers in a race where this was the greatest risk of loosing a 1-2 finish. When they give orders over the team radio that they want both cars to finish, I find it unlikely that they don't think about this when deciding the strategies as well.